How robust are tests for two independent samples?

Non-parametric procedures are sometimes in use even in cases where the corresponding parametric procedure is preferable. This is mainly due to the fact that in practical applications of statistical methods too much attention is paid to any violation of the normality assumption–normal distribution is, however, primarily supposed in order to easily derive the exact distribution of the statistic used within parametric approaches.As concerns the case of two independent samples and the comparison of (the expectations of) two populations the t-test and its non-parametric counterpart, the Wilcoxon (Mann–Whitney) test are of particular interest. These both serve as the illustrative example, given here.The Wilcoxon test compares the two distributions, indeed, but may in cases where we are interested in comparing expectations lead to significance even if the expectations are equal; this because any higher moments in the two populations may differ. On the other hand, the t-test is so robust against non-normality that there is nearly no need to use the Wilcoxon test in comparing expectations.Most results for continuous distributions have been obtained in a research group in Dummerstorf–Rostock some years ago and have been published by Herrendörfer [1980. Robustheit I, Arbeitsmaterial zum Forschungsthema Robustheit, Probleme der angewandten Statistik, vol. 4. Forschungszentrum Dummerstorf-Rostock, Heft], Herrendörfer et al. [1983. Robustness of statistical methods. II, Methods for the one-sample problem. Biometrical J. 25, 327–343], Rasch [1995. Mathematische Statistik. Johann Ambrosius Barth, Leipzig-Heidelberg], Rasch and Tiku [1984. Robustness of Statistical Methods and Nonparametric Statistics. VEB Deutscher Verlag der Wissenschaften, Berlin (D, Reidel Publ. Co., Dordrecht, Lancaster, Boston, Tokyo, 1985)], and Rasch and Guiard [2004. The robustness of parametric statistical methods. Psychol. Sci. 46(2), 175–208]. Most of the results are based on extensive simulation experiments with 10 000 runs each.In the present paper we discuss the two-sample problem firstly by summarising the results for continuous distributions from some preprints in German language; however, secondly we offer new results. All above we investigate the t-test's and the Wilcoxon test's robustness in cases of the distribution not being continuous.All the results are that in most practical cases the two-sample tt-test is so robust that it can be recommended in nearly all applications.